Unit 5, Lesson 9
Using Base-Ten Diagrams to Divide
Grade 6
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Elena used base-ten diagrams to find
She started by representing 372.
She made 3 groups, each with 1 hundred. Then, she put the tens and ones in each of the 3 groups. Here is her diagram for
Discuss with a partner:
Elena’s diagram for 372 has 7 tens. The one for
Where did the extra ones (small squares) come from?
Mai used base-ten diagrams to calculate
She then made 5 groups, each with 1 ten. There was 1 ten left. She decomposed it into 10 ones and distributed the ones across the 5 groups.
Here is Mai’s diagram for
Discuss these questions with a partner:
Mai should have a total of 12 ones, but her diagram shows only 10. Why?
She did not originally have tenths, but in her diagram each group has 4 tenths. Why?
What value has Mai found for
Four students share a $271 prize from a science competition. How much does each student get if the prize is shared equally? Show your reasoning.
To find
She placed 1 ten into each group, decomposed the remaining 1 ten into 10 ones, and went on distributing the units.
This diagram shows Elena’s initial placement of the units and the decomposition of 1 ten.
4 groups of base-ten diagrams. Group 1, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 2, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 3, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 4, 1 rectangle labeled, tens, 2 small rectangles labeled, tenths. Below the groups, one rectangle decomposed into 10 squares.
Here’s Elena’s finished diagram, showing the quotient of
Discuss with a partner:
One way to find the quotient of two numbers, such as
We can think of the division by 3 as splitting up 345 into 3 equal groups.
Each group has 1 hundred, 1 ten, and 5 ones, so
Base-ten diagrams can also help us think about division when the result is not a whole number. Let’s look at
4 groups of base-ten diagrams. Each group contains 2 rectangles labeled, tens, and 1 square labeled, ones. 2 squares are decomposed into 20 small rectangles labeled, tenths.
We can see that there are 4 groups of 21 in 86 with 2 ones left over. To find the quotient, we need to distribute the 2 ones into the 4 groups. To do this, we first need to decompose the 2 ones into 20 tenths and then put 5 tenths in each group.
Once the 20 tenths are distributed, each group will have 2 tens, 1 one, and 5 tenths, so
For some division problems, such as